Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Homological algebra for the representation Green functor for abelian groups

Author: Joana Ventura
Journal: Trans. Amer. Math. Soc. 357 (2005), 2253-2289
MSC (2000): Primary 55P91, 18G10
Published electronically: May 10, 2004
MathSciNet review: 2140440
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we compute some derived functors ${Ext}$ of the internal homomorphism functor in the category of modules over the representation Green functor. This internal homomorphism functor is the left adjoint of the box product.

When the group is a cyclic $2$-group, we construct a projective resolution of the module fixed point functor, and that allows a direct computation of the graded Green functor ${Ext}$.

When the group is $G=\mathbb{Z} /2\times\mathbb{Z} /2$, we can still build a projective resolution, but we do not have explicit formulas for the differentials. The resolution is built from long exact sequences of projective modules over the representation functor for the subgroups of $G$ by using exact functors between these categories of modules. This induces a filtration which gives a spectral sequence which converges to the desired ${Ext}$ functors.

References [Enhancements On Off] (What's this?)

  • 1. Serge Bouc.
    Green functors and ${G}$-sets.
    Springer-Verlag, Berlin, 1997. MR 99c:20010
  • 2. Andreas W. M. Dress.
    Contributions to the theory of induced representations.
    In Algebraic $K$-theory, II: ``Classical'' algebraic $K$-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pages 183-240. Lecture Notes in Math., Vol. 342. Springer, Berlin, 1973. MR 52:5787
  • 3. L. Gaunce Lewis, Jr.
    The box product of Mackey functors.
  • 4. L. Gaunce Lewis, Jr.
    The theory of Green functors.
    Unpublished notes, 1981.
  • 5. Saunders Mac Lane.
    Categories for the working mathematician.
    Springer-Verlag, New York, second edition, 1998. MR 2001j:18001
  • 6. John McCleary.
    A user's guide to spectral sequences.
    Cambridge University Press, Cambridge, second edition, 2001. MR 2002c:55027
  • 7. Jacques Thévenaz and Peter Webb.
    The structure of Mackey functors.
    Trans. Amer. Math. Soc., 347(6):1865-1961, 1995. MR 95i:20018
  • 8. Jacques Thévenaz and Peter Webb.
    Simple Mackey functors.
    In Proceedings of the Second International Group Theory Conference (Bressanone, 1989), number 23, pages 299-319, 1990. MR 91g:20011
  • 9. Charles A. Weibel.
    An introduction to homological algebra.
    Cambridge University Press, Cambridge, 1994. MR 95f:18001

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 55P91, 18G10

Retrieve articles in all journals with MSC (2000): 55P91, 18G10

Additional Information

Joana Ventura
Affiliation: Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

Received by editor(s): August 22, 2003
Published electronically: May 10, 2004
Additional Notes: The author was partially supported by FCT grant Praxis XXI/BD/11357/97 and a one year research grant from Calouste Gulbenkian Foundation
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society